Volume 23, Issue 1

The asymptotic normality of the total profitability and the number of purchase and sale transactions are established for a trade algorithm using a stationary m -dependent sequence of price increments. Approximate analytic dependences of their means and variances on the parameters of a price series and a trade algorithm are obtained. Parametric analysis of mathematical expectation for the number of purchase and sale transactions is performed for a trade algorithm with smoothing of the price series. The results of numerical experiments are presented.

A fast method (MFMA) for solution of the Poisson equation –Δ u = ƒ with homogeneous Dirichlet boundary conditions in a unit sector with an angle α , α > π , is proposed. As a result of application of MFMA under the condition , the solution accuracy of the order in the energy norm is achieved requiring O ( n 2 h –2) operations.

Existence and uniqueness of a weak solution for the system of primitive equations with stratification describing large-scale ocean dynamics, is proved ‘in the large’. The system is obtained from the original 3D primitive equations with viscosities depending on ρ z . The character of these dependences providing the existence and uniqueness of a solution ‘in the large’ is determined. More precisely, we consider that the viscosity coefficients in the vertical direction satisfy the following properties: they are non-increasing for ρ z ≤ 0 and constant for ρ z > 0 and satisfy the property: . Then it is proved that for an arbitrary time interval [0, T ], any viscosity coefficients in the horizontal direction, and any initial conditions , a weak solution exists and is unique and and the norms are continuous in t .

A method for estimating the parameters of a model describing the main specifics of tuberculosis transmission in Russia is constructed in the paper. The method presented here allows one to take into account the difference of social and economic characteristics in the regions and the quality of work of antituberculous centers. The dynamics of the main epidemiological indices under possible changes of both socioeconomic characteristics and the quality of work of antituberculous centers is simulated. The results obtained show that an improvement in economic conditions may positively influence the epidemic situation diminishing the prevalence of disease. Regions with high epidemiological indices are the most sensitive to changes.

This paper is concerned with a posteriori error estimates in terms of goal-oriented quantities for linear elliptic problems. The quantities are presented by locally supported linear functionals that serve as measures of the difference between the exact solution and its approximation in subdomains that can be selected using certain a priori knowledge on the expected properties of the solution. Typically, the evaluation of such type of error indicators is performed with the help of the so-called adjoint problem (see, e.g., [W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations . Birkhäuser, Berlin, 2003; R. Becker and R. Rannacher, Weighted a posteriori error control in finite element methods. In: Proc. of ENUMATH'97 , 1998, pp. 621–637]). In [S. Korotov, P. Neittaanmäki, and S. Repin, A posteriori error estimation of goal-oriented quantities by the superconvergent patch recovery. J. Numer. Math. (2003) 11 , No. 1, 33–59] an approach was suggested, in which the primal and adjoint problems are solved on non-coinciding meshes, and averaging of gradients is used to evaluate the term in the estimator that cannot be computed directly. However, superconvergence requires higher differentiability of the solutions and also regularity (quasi-uniformity) of both primal and dual meshes. In the present paper we propose a new method for computation of the estimator, where we use only superconvergence of the post-processed finite-element solution of the primal problem. At this point our analysis is based on the properties of the least square surface fitting operator investigated by J. Wang [J. Wang, A Superconvergence analysis for finite element solutions by the least-squares surface fitting on irregular meshes for smooth problems. J. Math. Study (2000) 33 , No. 3, 229–243]. We establish asymptotical properties of the new error indicator and verify its efficiency in a series of numerical tests.

It is proved that if f ( x ) is a polynomial of the degree not exceeding n with real coefficients, which satisfies the condition | f ( x )| ≤ 1 for all x ∈ [–1,1], then the sum of the absolute values of its coefficients attains its maximal value at f ( x ) = T n ( x ) = cos ( n arccos x ).

The paper is devoted to effective inversion of a saddle matrix corresponding to a discrete Laplace operator for a mixed problem.